6 research outputs found
Type-Two Well-Ordering Principles, Admissible Sets, and Pi^1_1-Comprehension
This thesis introduces a well-ordering principle of type two, which we call the Bachmann-Howard principle. The main result states that the Bachmann-Howard principle is equivalent to the existence of admissible sets and thus to Pi^1_1-comprehension. This solves a conjecture of Rathjen and Montalbán. The equivalence is interesting because it relates "concrete" notions from ordinal analysis to "abstract" notions from reverse mathematics and set theory.
A type-one well-ordering principle is a map T which transforms each well-order X into another well-order T[X]. If T is particularly uniform then it is called a dilator (due to Girard). Our Bachmann-Howard principle transforms each dilator T into a well-order BH(T). The latter is a certain kind of fixed-point: It comes with an "almost" monotone collapse theta:T[BH(T)]->BH(T) (we cannot expect
full monotonicity, since the order-type of T[X] may always exceed the order-type of X). The Bachmann-Howard principle asserts that such a collapsing structure exists. In fact we define three variants of this principle: They are equivalent but differ in the sense in which the order BH(T) is "computed".
On a technical level, our investigation involves the following achievements: a detailed discussion of primitive recursive set theory as a basis for set-theoretic reverse
mathematics; a formalization of dilators in weak set theories and second-order arithmetic; a functorial version of the constructible hierarchy; an approach to deduction chains (SchĂĽtte) and beta-completeness (Girard) in a set-theoretic context; and a beta-consistency proof for Kripke-Platek set theory.
Independently of the Bachmann-Howard principle, the thesis contains a series of results connected to slow consistency (introduced by S.-D. Friedman, Rathjen and Weiermann): We present a slow reflection statement and investigate its consistency strength, as well as its computational properties. Exploiting the latter, we show that instances of the Paris-Harrington principle can only have extremely long proofs in certain fragments of arithmetic
Well ordering principles for iterated 511-comprehension
We introduce ordinal collapsing principles that are inspired by proof theory but have
a set theoretic flavor. These principles are shown to be equivalent to iterated 1
1-
comprehension and the existence of admissible sets, over weak base theories. Our
work extends a previous result on the non-iterated case, which had been conjectured
in Montalbán’s “Open questions in reverse mathematics" (Bull Symb Log 17(3):431–
454, 2011). This previous result has already been applied to the reverse mathematics
of combinatorial and set theoretic principles. The present paper is a significant contribution to a general approach that connects these fields
Applicazioni di analisi ordinale: il teorema di Kruskal.
Si fornisce una caratterizzazione del teorema di Kruskal con la tecnica dell'analisi ordinale, uno dei principali strumenti della teoria della dimostrazione
Two applications of analytic functors
AbstractWe apply the theory of analytic functors to two topics related to theoretical computer science. One is a mathematical foundation of certain syntactic well-quasi-orders and well-orders appearing in graph theory, the theory of term rewriting systems, and proof theory. The other is a new verification of the Lagrange–Good inversion formula using several ideas appearing in semantics of lambda calculi, especially the relation between categorical traces and fixpoint operators